Prof. Dr. Markus Bachmayr (Universität Mainz)Stability of Low-Rank Tensor Representations and
Structured Multilevel Preconditioning for Elliptic PDEs
Folding grid-value vectors into high-order tensors in combination with
low-rank representation in the tensor-train format leads to highly
efficient approximations for various classes of functions. These include
solutions of elliptic PDEs on nonsmooth domains or with oscillatory data,
for which simple discretizations parametrized by low-rank tensors have
been shown to result in highly compressed, adaptive approximations.
Straightforward choices of the underlying basis, such as piecewise
multilinear finite elements on uniform tensor product grids, lead to the
well-known matrix ill-conditioning of discretized operators. We
demonstrate that for low-rank representations, the use of tensor structure
can additionally lead to representation ill-conditioning, a new effect
specific to computations in tensor networks. We then show that the issues
of matrix ill-conditioning and representation ill-conditioning can be
circumvented simultaneously by combining classical ideas of multilevel
preconditioning and more recent techniques for the low-rank representation
of matrices. Specifically, we construct an explicit tensor-structured
representation of a BPX preconditioner with ranks independent of the
number of discretization levels. Combined with a carefully constructed
representation of differentiation and L2-projection, it allows to avoid
both matrix and representation ill-conditioning. Numerical tests,
including problems with highly oscillatory coefficients, show that our
result paves the way to reliable and efficient solvers that remain stable
for mesh sizes near machine precision with up to 1015 nodes in each
dimension.
This is joint work with Vladimir Kazeev.

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

M. Sc. Lukas Sawatzki (Universität Marburg)Coorbit Theorie und ihr Kern-Problem
Since the 1980's the coorbit theory provides a unified approach to certain function
spaces and atomic decompositions.
At the heart of this theory we always find a transform acting on L2-functions,
e.g. the wavelet, shearlet or Gabor transform.
These transforms always have an underlying group structure and are strongly
connected to square-integrable representations of these groups.
With the help of these transforms we can now define associated smoothness spaces,
where smoothness is measured by the decay properties of the so-called voice
transform of given functions.
In this manner we obtain a unified approach to define function spaces suitable for
certain transforms.
Additionally, by discretizing the representation, we can find atomic decompositions
and Banach frames for these spaces.
An application of the coorbit theory to the wavelet transform gives us, e.g., the
well-known homogeneous Besov spaces.
In the coorbit theory an integral kernel, also called reproducing kernel, plays a
fundamental role, and in the classical case it is assumed to be integrable.
Unfortunately, this assumption is not always fulfilled.
It is nevertheless possible to define meaningful coorbit spaces also in this setting.
However, the atomic decomposition of these spaces and the construction of Banach
frames call for additional assumptions on the integral kernel.

17:00 Uhr:

Prof. Dr. Michael Möller (Universität Siegen)Optimization Problems in Machine Learning Applications
Machine learning algorithms, most prominently techniques under the name of
deep learning, are currently dominating many computer vision benchmarks.
The success of such methods largely depends on the ability to train the
desired network architectures, i.e. optimize the sum over millions of cost
functions each of which is a deeply nested function of the desired
parameters to optimize for. In this talk I will introduce some of our
research on the efficient solution of such optimization problems including
recent work on bi-level optimization problems.

Veranstaltungen im Sommersemester 2018

Prof. Dr. Claudia Schillings (Universität Mannheim)Uncertainty Quantification for Inverse Problems
Uncertainty Quantification is an interesting, fast growing research area
aimed at developing methods to
address the impact of parameter, data and model uncertainty in complex
systems. In this talk we will
focus on the identification of parameters through observations of the
response of the system - the
inverse problem. The uncertainty in the solution of the inverse problem
will be described via the
Bayesian approach. In cases, where the model evaluations are prohibitively
expensive, ad hoc methodssuch as the Ensemble Kalman Filter (EnKF) for inverse problems are widely
and successfully used by
practitioners in order to approximate the solution of the Bayesian
problem. The low computational costs, the straightforward implementation and their
non-intrusive nature make them
appealing in various areas of application, but, on the downside, they are
underpinned by very limited
theoretical understanding. In this talk, we will discuss an analysis of
the EnKF based on the continuous
time scaling limits, which allows to derive estimates on the long-time
behaviour of the EnKF and, hence,
provides insights into the convergence properties of the algorithm.

16:00 Uhr:

Tee/Kaffee

16:30 Uhr:

Prof. Dr. Markus Hansen (Universität Marburg)Inferring Interaction Rules from Observations of Evolutive Systems
In this talk we are concerned with the learnability of nonlocal interaction kernels for first order systems modeling certain
social interactions, from observations of realizations of their dynamics. In particular, we assume here that the kernel to be learned
is bounded and locally Lipschitz continuous and that the initial conditions of the systems are drawn identically and independently at
random according to a given initial probability distribution. Then the minimization over a rather arbitrary sequence of (finite dimensional) subspaces of a least square functional measuring the discrepancy from
observed trajectories produces uniform approximations to the kernel on compact sets. The convergence and its rate are obtained by combining
mean-field limits, transport methods, and a Γ-convergence argument. A crucial condition for the learnability is a certain coercivity property of the least square functional, majoring an L2-norm discrepancy to the kernel with respect to a probability measure, depending on the given initial probability distribution by suitable push forwards and transport maps. We illustrate the convergence result by means of several numerical experiments.

17:15 Uhr:

Dr. Kersten Schmidt (TU Darmstadt)A High Order Galerkin Method for the Approximation of Contour Integrals
We introduce a novel method to compute approximations of contour integrals
in some bounded domain in Ω⊂ℝd. The new method is
based on the coarea formula in combination with a Galerkin projection. As
such it fits seamlessly into the spirit of hp/spectral finite element
methods and circumvents the expensive and technical computation of contours.
Only integrals over the domain Ω have to be evaluated. We provide
convergence estimates showing that a high order convergence can be achieved
provided that the data is sufficiently smooth. Moreover, we analyze the
contribution to the discretization error if an approximated level set
function , e.g. its Galerkin approximation, is used. The theoretical results
are supplemented by numerical experiments and an example application for
plasma modeling in nuclear fusion.

Veranstaltungen im Wintersemester 2017/18

Dr. Thomas Batard (TU Kaiserslautern) Geometric Variational Models for Color Images Correction
Due to physical and technological limitations of the acquisition process of a real-world scene by a digital camera, the
output image of the camera processing pipeline is a degraded version of the observed scene. In this talk, I will present some mathematical models for image processing whose aim is to correct the output image in
order to make it perceptually closer to the observed scene.

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

M.Sc. Philipp Keding (Universität Marburg)Quarklet Frames in Adaptive Numerical Schemes
This talk is concerned with new discretization methods for the numerical
treatment of elliptic partial differential equations. We derive an
adaptive frame scheme that is based on quarkonial decompositions. These
new frames are constructed from a finite set of functions by translation,
dilation and multiplication by monomials. By means of nonoverlapping
domain decompositions, we establish quarkonial frames on domains that can
be decomposed into the union of parametric images of unit cubes. We also
show that these new representation systems constitute stable frames in
scales of Sobolev spaces. The construction is performed in such a way
that, similar to the wavelet setting, the frame elements, the so-called
quarklets, possess a certain amount of vanishing moments. This enables us
to generalize the basic building blocks of adaptive wavelet algorithms to
the quarklet case. The applicability of the new approach is demonstrated
by numerical experiments for the Poisson equation on L-shaped domains.

17:00 Uhr:

Priv.-Doz. Dr. Michael Gnewuch (Universität Kiel)Infinite-Dimensional Integration
Integrals over functions with an infinite number of variables appear in
applications such as molecular chemistry, physics or quantitative finance.
Complex stochastic models, e.g., are often based on a sequence of
independent and identically distributed random variables, implying that
expectations can be represented as infinte-dimensional integrals.
In the talk we want to discuss how to approximate infinite-dimensional
integrals. We will, e.g., consider integrands that belong to weighted
Sobolev-spaces of dominated mixed smoothness.
Optimal algorithms can be constructed as follows: We start with optimal
algorithms for univariate integration and use them as building blocks
for Smolyak algorithms (aka sparse grid methods) for multivariate
integration which in turn are used as building blocks of so-called
multivariate decomposition methods for infinite-dimensional integration.
With the help of these algorithms we can, in particular, establish the
following result: For classes of 'sufficiently smooth' integrands
the infinite-dimensional integration problem is (essentially) not harder
than the corresponding univariate (i.e., the 'one-dimensional')
integration problem.

Dr. Mirjam Walloth (TU Darmstadt)A-posteriori error estimators for the Signorini problem
The talk deals with the adaptive numerical simulation of
contact problems based on
residual-type a posteriori estimators. The estimators are easy to
compute and provably reliable,
efficient and localized. The latter properties enable a good
resolution of the free boundary
while avoiding over-refinement in the active set of contact. We
consider continuous as well as
discontinuous finite elements for the numerical simulation of static
and time-dependent contact
problems.

Dr. Daniel Rudolf (Friedrich-Schiller-Universität Jena)Perturbation theory for Markov chains
By using perturbation theory for Markov chains we derive explicit estimates of the bias of an approximate version of a geometrically ergodic Markov chain.
We apply this result to a noisy Metropolis-Hastings algorithm and discuss also some consequences for the integration error of such Markov chain Monte Carlo methods.

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

Kerstin Lux (Universität Mannheim)Simulation studies on stochastic differential equations with discontinuous drift coefficient
The Euler-Maruyama scheme (EMS) is one of the standard schemes to obtain numerical approximations of stochastic differential equations (SDEs). Its
convergence properties are well-known in the case of Lipschitz-continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior which results in discontinuous coefficients of the corresponding SDE. In this talk, we will analyze numerical convergence properties of the EMS as well as of an explicit order 1.5 strong scheme due to Platen for SDEs with
a piecewise constant drift coefficient and a constant diffusion coefficient. This type of SDEs arises in some rank-based stock market models. Therefore, as
an application of our numerical analysis, we will give numerical results on the long-term ranking behavior within a stock market.
This is joint work together with Simone Göttlich and Andreas Neuenkirch.

17:00 Uhr:

Prof. Dr. Oleg Davydov (Justus-Liebig-Universität Gießen)Meshless Finite Difference Methods
After a brief discussion of the motivations and some history of the generalized finite difference methods, we concentrate on their recent meshless versions relying on kernel based numerical differentiation on
irregular centers. In particular, recent consistency estimates and adaptive algorithms for elliptic equations will be discussed.

Prof. Dr. Reinhold Schneider (TU Berlin)Convergence rates of basis sets in Density Functional Theory and Hartree-Fock for electronic structure calculation.
The numerical solution of the stationary electronic Schroedinger equation is a fundamental task in
the numerical simulation of atomic and molecular behaviour, with widespread applications in
chemistry, molecular biology, solid state physics and material sciences. In Hartree Fock, resp. Density
Functional Theory (DFT) the highdimensional linear Schroedinger equation is replaced by a system of
nonlinear but low dimensional Hartree Fock resp. Kohn Sham equations. This allows the treatment of
relativley large systems. We will discuss the numerical treatment of DFT as a constrained
optimization problem. We are considering LAPW, numerical atomic orbitals and Gaussian basis
functions and hp-finite elements for all electron calculations. We have shown that, under mild
assumptions, these functions are converging almost exponentially, i.e. with any algebraic
convergence rate. These results may explain the success of these functions in chemical and physical
applications.
This is joint work together with M. Bachmayr (RWTH Aachen) and H. Chen (U Warwick).

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

Esther Hans (Johannes Gutenberg-Universität Mainz)Globally convergent multilevel B-semismooth Newton methods for l1-Tikhonov regularization.
Tikhonov regularization with l1 coefficient penalties is widely used for the regularization of ill-posed
problems if the underlying solution is sparse with respect to some given basis. Due to their local
convergence speed, semismooth Newton methods are competitive for the minimization of the
resulting nonsmooth Tikhonov functional. In the first part of the talk, we are concerned with the
globalization of existing locally superlinearly convergent semismooth Newton methods. Here, we
present a damped generalized Newton method based on the B(ouligand)-derivative of the specific
nonlinearity. The second part of the talk treats an acceleration scheme combining the resulting
globally convergent generalized Newton method with algebraic multilevel methods, recently
introduced by Treister, Turek and Yavneh.
The results are based on joint work with Thorsten Raasch.

17:00 Uhr:

Dr. Mario Hefter (TU Kaiserslautern)Optimal strong approximation of the one-dimensional squared Bessel process.
We consider a stochastic differential equation (SDE) describing a one-dimensional squared Bessel
process and study strong (pathwise) approximation of the solution at the final time point t=1. This
SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process where the boundary point zero is
accessible. We consider numerical methods that have access to values of the driving Brownian
motion at a finite number of time points. We show that the polynomial convergence rate of the n-th
minimal errors for the class of adaptive algorithms as well as for the class of algorithms that rely on
equidistant grids are equal to infinity and 1/2, respectively. As a consequence, we obtain that the
parameters appearing in the CIR process affect the convergence rate of strong approximation.
A key step in the proofs consists of identifying the pathwise solution of the SDE and link this problem
to global optimization under the Wiener measure.
This is joint work with James M. Calvin and Andre Herzwurm.

Prof. Mike Giles (University of Oxford)Multilevel Monte Carlo for reduced accuracy computations
Motivated by FPGAs and GPUs which are capable of performing computations
with different levels of precision at different speeds, we consider
Multilevel Monte Carlo simulations for SDEs with varying accuracy on
different levels. This includes both the accuracy of individual
arithmetic operations due to the roundoff error, and the accuracy in the
approximation of intrinsinc functions such as the inverse error function
or the inverse Normal CDF which can be used to convert uniform random
variables into Normal random variables.

The emphasis in the presentation will be on the construction of valid
multilevel approximations, and their mathematical modeling and analysis,
but we hope to have some numerical results as well -- this will be work
to be completed during the two weeks before the talk!

Co-authors: Klaus Ritter, Mario Hefter, Steffen Omland

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

Prof. Dr. Oliver Kolb (Univ. Mannheim)Modeling, simulation and optimization on networks.
In the main part of this talk, we consider several applications on networks within a unified simulation and optimization framework, in particular gas and water supply networks, road traffic and production systems. Based on the underlying macroscopic models, we first apply appropriate discretization schemes to each problem. To solve optimization tasks, we use an SQP solver and compute gradient information with a first-discretize adjoint approach. The latter information can additionally be exploited for error estimation and an adaptive simulation/optimization algorithm. Various numerical results are shown to demonstrate the applicability of the presented framework. Finally, if time permits, some recent results on WENO discretization schemes will be discussed.

17:00 Uhr:

Dipl-Math. Dominik Garmatter (Goethe-Universität Frankfurt)The reduced basis method and its application to inverse problems.
The numerical solution of nonlinear inverse problems such as the identification of a parameter in
a partial differential equation (PDE) from a noisy solution of the PDE via iterative regularization
methods usually requires numerous amounts of forward solutions of the respective PDE. Since this can
be very time-consuming, it is highly desirable to speed up the solution process.

The reduced basis method is a model order reduction technique that can yield a significant decrease
in the computational time of the PDE solution, especially in a many-query context as it is the case in
the above situation.

The first part of this talk will provide an introduction to the reduced basis method for elliptic,
coercive problems. The second part will deal with the task of combining the reduced basis method
with iterative regularization algorithms for ill-posed inverse problems in order to redcue their overall
computational time. The main objective will be the development of the new Reduced Basis Landweber
method. Numerical results will fortify the approach.

Veranstaltungen im Sommersemester 2015

Prof. Dr. Bastian von Harrach (Universität Stuttgart)Inverse problems and medical imaging
Medical diagnosis has been revolutionized by noninvasive imaging methods such as computerized tomography (CT) and magnetic resonance
imaging (MRI). These great technologies are based on mathematics. If the patient's interior was known then we could numerically simulate the outcome of physical measurements performed on the patient. Medical imaging requires solving the corresponding inverse problem of determining the patient's interior from the performed measurements. In this talk, we will give an introduction to inverse problems in medical imaging, and discuss the mathematical challenges in newly emerging techniques such as electrical impedance tomography (EIT), where electrical currents are driven through a patient to image its interior. EIT leads to the inverse problem of determining the coefficient in a partial differential equation from (partial) knowledge of its solutions. We will describe recent mathematical advances on this problem that are based on monotonicity relations with respect to matrix definiteness and the concept
of localized potentials.

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

Dr. Ronny Bergmann (TU Kaiserslautern)A Second Order Non-smooth Variational Model for Restoring Manifold-valued Images.
In many real world situations, measured data is noisy and nonlinear, i.e., the data is given as values in a certain manifold.
Examples are InSAR images and the hue channel of HSV, where the entries are phase-valued, directions in R^n, which are data given on S^{n-1}, and diffusion tensor magnetic resonance imaging, where the obtained data items are symmetric positive definite matrices.

In this talk we extend the recently introduced total variation model on manifolds by a second order TV type model. We first introduce second order differences on manifolds in a sound way using the induced metric on Riemannian manifolds. By avoiding a definition involving tangent bundles, this definition allows for a minimization employing the inexact cyclic proximal point algorithm, where the proximal maps can be computed using Jacobian fields. The algorithm is then applied to several examples on the aforementioned manifolds to illustrate the efficiency of the algorithm.

17:00 Uhr:

Prof. Dr. Christoph Erath (TU Darmstadt)Adaptive Coupling of Finite Volume and Boundary Element Methods
In many fluid dynamics problems the boundary conditions may be unknown, or the domain may be unbounded. Also mass conservation and stability with respect to dominating convection is substantial. Therefore, we introduce a (rather) new discrete scheme to address these issues. More precisely, we couple the finite volume method and the boundary element method and provide some theoretical results.
Also robust a posteriori estimates are developed and analysed, which allow us to use an adaptive mesh-refinement algorithm. This strategy turns out to be very suitable for the numerical treatment of transmission problems, which have singularities or boundary/internal layers. Several numerical examples illustrate the analytical results and the effectiveness of the new conservative adaptive coupling method.

Martin Altmayer (Universität Mannheim)Quadrature of Discontinuous SDE Functionals Using Malliavin Integration By Parts
The Heston Model is a popular stochastic volatility model in mathematical finance. In its classical form the volatility process is given by a CIR process, whereas in the generalized form the volatility
follows a mean-reverting CEV process.

While there exist several numerical methods to compute functionals of the Heston price, the convergence order is typically low for discontinuous functionals. In this talk, we will study an approach based
on the integration by parts formula from Malliavin calculus to overcome this problem: The original function is replaced by a function involving its antiderivative and by a Malliavin weight. Using the drift-implicit Euler scheme for the square root of the volatility, we will construct an estimator for which we can prove that it has L2-convergence order 1/2 even for discontinuous functionals. This leads to an efficient multilevel algorithm, also in the multidimensional case.

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

Dr. Markus Weimar (Univ. Marburg)On Optimal Wavelet Approximations in Spaces of Besov Type
This talk is concerned with the approximation of embeddings between Besov-type spaces defined on bounded multidimensional domains or (patchwise smooth) manifolds. We compare the quality of approximation of
three different strategies based on wavelet expansions. For this purpose, sharp rates of convergence corresponding to classical uniform refinement, best $N$-term, and best $N$-term tree approximation will
be presented. In particular, we will see that whenever the embedding of interest is compact greedy tree approximation schemes are as powerful as abstract best $N$-term approximation and that (for a large range of parameters) they can outperform uniform schemes based on a priori fixed (hence non-adaptively chosen) subspaces. This observation justifies the use of adaptive non-linear algorithms in computational practice, e.g., for the approximate solution of boundary integral equations arising from physical applications.

If time permits, implications for the related concept of approximation spaces associated to the three approximation strategies will be discussed.

The results to be presented are work in progress within the framework of the DFG-Project Adaptive Wavelet and Frame Techniques for Acoustic BEM" (DA 360/19-1).

Dr. Martin Gutting (Universität Siegen)Multiscale Analysis on the Earth's Surface Accelerated by the Fast Multipole Algorithm
By the use of an underlying Runge sphere harmonic scaling functions and wavelets can be constructed
on regular surfaces such as the surface of the Earth. They allow a space{frequency decomposition of
geophysical quantities on the surface. Moreover, due to their localizing properties regional modeling
or the improvement of a global model is possible. The acceleration of the convolution by the fast multipole method is possible for certain types of
harmonic scaling functions and wavelets. The main idea of the fast multipole algorithm consists of a
hierarchical decomposition of the computational domain into cubes and a kernel approximation for
the more distant points. The kernel evaluation is performed directly only for points in neighboring
cubes on the finest level. The contributions of the other points are transferred into a set of coeffcients. The kernel approximation is applied on the coarsest possible level using translations of these coefficients. This reduces the numerical effort of the convolution for a prescribed accuracy of the
kernel approximation. Multiscale methods are known to possess a tree algorithm that allows the computation of the lower
frequency scales from a starting scale that contains the highest frequency parts of the signal. The
application of the fast multipole method can accelerate the computation of this starting point as well
as the tree algorithm itself. Applications to gravitational field modeling are presented. Finally, the extension to boundary value
problems is considered where the boundary is the known surface of the Earth itself.

15:45 Uhr:

Tee/Kaffee

16:15 Uhr:

Dipl.-Math. Max Nattermann (Univ. Marburg)Robust Optimal Design of Experiments and a Higher Order Sensitivity
Analysis of Parameter Estimation Problems
When dealing with the task of estimating parameters by the use of a set of noisy data, the number of available
measurements is limited. Therefore, in optimum experimental design it is tried to identify the system settings
with those measurements which allow the most reliable estimate.
In this talk we are going to present properties and examples of a new and robust objective function of optimum
experimental design, which is based on a higher order sensitivity analysis of the underlying parameter estimation
problem.

17:00 Uhr:

Dr. Nikolaos Sfakianakis (Johannes Gutenberg-Universität Mainz)Mathematical modeling and numerical simulation of cancer dynamics
One of the primer aims in cancer research is to understand the way
cancer cells interact with their environment, the dynamics and phenomena
they develop, the way they react and adjust to external stimuli, how
they move, proliferate, and how they metastasize. This aim is highly
interdisciplinary involving medical, biological, and mathematical
components. In this direction, our research effort focuses on the
mathematical modeling and the numerical simulations of the first step of
tumour dynamics: the invasion of the Extracellular Matrix.
The models that we work on are Advection-Reaction-Diffusion systems,
where the motility of the cells is dominated by the advection/taxis
dynamics. We include in the models Cancer Stem Cells as well as the
dynamical transitions between differentiated and cancer stem cells. We
study the consequences of heterogeneities -caused by the presence of
both cancer cell types- on the invasion of the extracellular matrix. To
this end we employ a plethora of numerical methods adjusted to fit the
needs of such systems. The dynamics though of the solutions are quite
rich and very fine numerical grids are necessary to consistently resolve
the dynamics. To alleviate the numerical burden of very fine grids we
use mesh refinement techniques to increase the grid resolution only
locally.
Joint work with:
M. Lukacova, N. Hellmann, and N. Kolbe
Thanks:
"Alexander von Humboldt Foundation" and the "Center for Computational
Sciences in Mainz".

Veranstaltungen im Sommersemester 2012

Prof. Dr. Kasso Okoudjou (University of Maryland, Department of Mathematics)Probabilistic frames
In the first part of this talk I will give an overview of
finite frame theory from a probabilistic point of view. In
particular, I will review the notion of probabilistic frames and
indicate how it is a natural generalization of frames. In fact,
probabilistic frames appear in many other areas such as statistics,
convex geometry, the theory of spherical design, and quantum
computing. The second part of the talk will focus on counter parts to
concepts such as tight frames, frame potentials in the setting of
probabilistic frames. The talk is based on recent joint work with
Martin Ehler.

Dr. Janosch Rieger (Goethe-Universität Frankfurt am Main, FB Mathematik)Uncertainty quantification and partial differential inclusions
Uncertainty quantification is an area of increasing practical importance. In
some applications, it is desirable to understand the distribution of the solu-
tion of an operator equation or a partial differential equation provided that
the right-hand side is a random variable with known distribution. This type
of uncertainty is currently an object of intense research.
A very different type of problem arises if only bounds and no distributions
are known for the right-hand sides. The elliptic partial differential inclusion
Au \in F(u) in \Omega; u = 0 on \partial\Omega (+)
considered in this talk models deterministic uncertainty and constrained con-
trol problems. We currently try to develop the necessary analytical back-
ground and numerical methods for an effcient approximation of the set of
all solutions of (+).
First experiments in the linear elliptic case show that it is difficult to
obtain good results by discretizing the multivalued right-hand side F. It is
much better to project inclusion (+) to some finite-dimensional space and
approximate the solution set of the resulting algebraic inclusion.
The semi-linear elliptic case is much more involved. Set-valued Nemytskii
operators have to be considered for the projection of (+) to a finite element
space. In order to guarantee uniform convergence of the Galerkin solution
sets, the so-called relaxed one-sided Lipschitz property is imposed on the
right-hand side F, which is a generalization of the classical OSL property.
Under this assumption, it is also possible to discretize and approximate the
unknown Galerkin solution sets.

Veranstaltungen im Wintersemester 2010/2011

Prof. Dr. Steffen Dereich (Universität Marburg)Multilevel Monte-Carlo algorithms for Lévy-driven SDE's with Gaussian correction
In this talk we analyze multilevel Monte-Carlo algorithms for the
computation of Ef(Y), where Y=(Yt)t \in [0,1] is the
solution of a Lévy-driven SDE and f is a real-valued function on the
path space.
We discuss several approaches and prove upper bounds for the worst case
error inferred on the class of Lipschitz continuous functionals (w.r.t.
supremum norm). Here, the dominant term of the upper estimate can be
expressed in terms of the Blumenthal-Getoor index.
Comparing the approaches, we find a significant improvement in the error
estimates when applying a Gaussian correction for the small jumps in the
case where the Blumenthal-Getoor index is larger than one.
Our analysis is very robust in the sense that we do not impose
particular assumptions on the structure of the Lévy process (e.g.
subordinated Lévy processes) except the existence of second moments.

Prof. Dr. Klaus Böhmer (Universität Marburg)A Nonlinear Discretization Theory with Applications to Meshfree Methods: Quasilinear and Fully Nonlinear PDEs
We extend for the first time the linear discretization theory of Schaback,
developed for meshfree methods, to nonlinear operator equations,
relying heavily on methods of Böhmer, Vol I. There is no restriction to
elliptic problems or to symmetric numerical methods like Galerkin techniques.
Trial spaces can be arbitrary, but have to approximate the solution
well, and testing can be weak or strong. We present Galerkin techniques
as an example. On the downside, stability is not easy to prove for special
applications, and numerical methods have to be formulated as optimization
problems. Results of this discretization theory cover error bounds
and convergence rates. These results remain valid for the general case
of quasilinear and fully nonlinear elliptic differential equations of second
order.

Mehdi Slassi (TU Darmstadt)The uniform free-knot spline approximation of Stochastic Differential Equations
We analyze the pathwise approximation of scalar stochastic differential equations (SDE) by polynomial splines with free knots. The pathwise distance between the solution and its approximation is measured globally on the unit interval in the \(L_{\infty}\)-norm, and the expectation of this distance is of concern here. We introduce a numerical method \(\widehat{X}_{k}\) with \(k\) free knots which is based on asymptotic optimal approximation of a scalar Brownian motion by splines with free knots. For general SDEs, we establish an upper bound of order \(1/\sqrt{k}\) with an explicit asymptotic constant for the approximation error of \(\widehat{X}_{k}\). In particular case of SDEs with additive noise this asymptotic upper bound is sharp.

17:00 Uhr:

Prof. Dr. Maria Lukacova (Universität Mainz)Finite Volume Evolution Galerkin Schemes (theory & applications in geophysical flow)
We present a newly developed well-balanced FV evolution Galerkin scheme for multidimensional systems of hyperbolic conservation laws. These methods are based on the theory of bicharacteristics and take all infinitely many directions of wave propagation into account. A typical characteristic of geophysical flows is their multiscale behaviour with wave speeds differing by orders of magnitude. Thus, the gravitational waves are much faster than advection waves. To alleviate a sever CFL stability condition and approximate efficiently low Froude number flows we have developed a large time step variant of the FVEG method. The behaviour of schemes will be illustrated by numerical experiments.